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This is my version of explanation. I would suggest you to come up with your own explanation. The best way would be for you to try explain this to somebody else in your own words.

Following is my version of explanation, but this is just an example. You may come up with a better version.

Example of Vector Field - Trajectory - Damped Oscillation

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

v = -1:0.2:1;

[y1,y2] = meshgrid(v);

u1=y2;

u2=-y1 - 0.25 .* y2;

function dy_dt = f(y, t)

  dy_dt = zeros (2,1);

  dy_dt(1) = y(2);

  dy_dt(2) = -y(1) - 0.25 .* y(2);

endfunction

y0 = [0.0;1.0];

t = 0:0.2:2.0;

y = lsode ("f", y0, t);

tStr = sprintf("t = %02.1f",t(end));

hFig = figure(1);

subplot(1,2,1);

plot(t,y(:,1),'r-',t,y(:,2),'b-');

xlabel('time');

xlim([0 25]);

ylim([-1 1]);

legend('y2(t)','y1(t)');

title(tStr);

subplot(1,2,2);

hold on;

quiver(y1,y2,u1,u2);

box on;

plot(y(:,1),y(:,2),'r-'); xlabel('y(2)');ylabel('y(1)');

axis([-1 1 -1 1]);

plot(y0(1),y0(2),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',10);

plot(y(end,1),y(end,2),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',10);

title(tStr);

hold off;

set(hFig,'Position',[300 300 780 300]);